Extended precedence theorems for general E 13

becomes a necessary condition for the dominance of (i, j). Further-more, the precedence theorems that were discussed in Section 2.2.1 may not be applicable as is. Consider the example depicted in Figures 2.1-2.2, and Table 2.1 with E = {(1, 3), (2, 4)} and S = E.

We investigate an additional precedence constraint from job 3

14 2.2. PRECEDENCE THEOREMS

1

2 3

4

Figure 2.2: A project instance: Transitive closure of E ∪ {(3, 2)}

From: job j M job i

To: β_i job i γij job j α_j

Figure 2.3: Swap strategy when E 6= ∅

to job 2. Since I(E, 3, 2) = 1 (based on K1, K4 and K5), we add the pair (3, 2) to S. As depicted in Figure 2.2, (3, 2) implies the transitive edges (1, 2), (1, 4) and (3, 4). Thus, we end up with the sequence s¹ = (1, 3, 2, 4) with T (s¹) = 159, while for the optimal sequence s^∗ = (2, 4, 1, 3), T (s^∗) = 119. The two transitive edges (1, 2) and (1, 4) are not dominant, and consequently remove the optimal solutions. This counterexample shows that with general E, Kanet’s and Emmons’ conditions cannot be directly invoked, i.e., I(S, i, j) = 1 is not sufficient to conclude (i, j) ∈ D. Hence, if E 6= ∅ then C(S) is not necessarily a subset of D.

Kanet (2007) uses “swap” and “insert-after” strategies to prove his dominance theorems. Conditions E2–3 and K4–7 are obtained via the insert-after strategy, while Conditions E1 and K1–3 are derived using the swap strategy. Condition K1 generalizes E1, K4 and K5 generalize E2, K7 is the same as E3, and K2, K3 and K6 are entirely new in the sense that they can lead to the conclusion that a pair (i, j) ∈ D even when w_i< w_j.

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TARDINESS PROBLEM 15

From: job j M job i

To: M \ α_j job i job j αj

Figure 2.4: Insert-after strategy when E 6= ∅

An illustration of the swap and insert-after strategies for gen-eral E and a given dominant S is provided in Figures 2.3 and 2.4, where β_i = (M ∩ B_i^E), α_j = (M ∩ A^E_j ) and γ_ij = M \ (α_j ∪ β_i).

The symbol M represents the set of intermediate jobs between i and j. “From” representsany sequence that respects S, and “To” is the resulting sequence after swapping j and i or inserting j after i.

The latter sequence respects E but not necessarily S, i.e., a number of dominant precedence constraints in S \ E might be violated. A sufficient condition for the dominance of (i, j) has the structure

LB(TI(i)) ≥ UB(TD(j)) + UB(TD(γ_ij)) + UB(TD(α_j)), (2.1) where LB(·) and UB(·) are lower and upper bound functions, re-spectively, TI(i) is the tardiness improvement of job i, and TD(i) the tardiness degradation. Note that TD(β_i) = 0.

If an activity pair (i, j) satisfies Condition (2.1) forevery feas-ible M and G(N, E ∪ {(i, j)}) is acyclic, then if j precedes i in a given schedule, we can exchange the two jobs without increasing the tardiness function. Thus, for an acyclic set of activity pairs {(i, j), (k, l), . . . } that each satisfy Condition (2.1), any optimal schedule that is not compatible with one or more of these pairs cannot be harmed by making as many interchanges as necessary to obtain an optimal schedule that respects all the pairs. Therefore,

16 2.2. PRECEDENCE THEOREMS

if the “To” sequence does not respect S, then by a finite number of swaps and insert-afters, it can be transformed into a sequence that respects S, such that the final sequence is at least as good as the intermediate sequences. Given an instance G(N, E), let V ⊆ D be the set of all activity pairs that satisfy Condition (2.1). We conclude:

Proposition 2.2. Any S ⊇ E for which (S \ E) ⊆ V is dominant iff S is acyclic.

Hence, we search for an inclusion-maximal acyclic S ⊇ E such that (S \ E) ⊆ V .

In Condition (2.1), the completion times of jobs i and j depend on P (α_j) and P (β_i), so TI(i) and TD(j) depend on β_iand α_j. Also, TD(α_j) can be positive in both strategies. Finally, even if p_i≤ p_j, the value TD(γ_ij) can still be positive in the swap strategy. We therefore extend Emmons’ and Kanet’s theorems under the extra requirement that α_j = β_i = ∅.

Proposition 2.3. A^E_j ⊆ A^S_i is a sufficient condition for αj = ∅. Proof. Proof. Remember that αj = M ∩ A^E_j . The requirement that α_j is empty means all jobs in A^E_j are scheduled after job i.

Intuitively, the condition A^E_j ⊆ A^E_i is sufficient to ensure α_j = ∅.

Since the “From” sequence is feasible not only to E but also to S, the condition A^E_j ⊆ A^S_i is also sufficient.

Analogously, we can prove:

Proposition 2.4. B_i^E ⊆ B_j^S is a sufficient condition for βi = ∅. The conditions A^E_j ⊆ A^E_i , A^S_j ⊆ A^E_i and A^S_j ⊆ A^S_i are also sufficient for α_j = ∅ and B_i^E ⊆ B_j^E, B_i^S ⊆ B_j^E and B^S_i ⊆ B_j^S are

CHAPTER 2. THE SINGLE-MACHINE WEIGHTED

TARDINESS PROBLEM 17

also sufficient for β_i= ∅, but A^E_j ⊆ A^S_i and B_i^E ⊆ B_j^S are easier to fulfill compared to the other alternatives, since E ⊆ S.

When α_j = β_i = ∅, a sufficient condition for the dominance of (i, j) takes the form

LB(TI(i)) ≥ UB(TD(j)) + UB(TD(M )). (2.2) Kanet (2007) proves that given a dominant S and (i, j) ∈ N × N , if I(S, i, j) = 1 then (i, j) satisfies Condition (2.2). We summarize our findings with the following formal statements.

Proposition 2.5. When E 6= ∅ then each of Conditions E1 and K1–3 together with A^Ej ⊆ A^S_i and Bi^E ⊆ B_j^S imply (i, j) ∈ D.

Proposition 2.6. When E 6= ∅ then each of Conditions E2–3 and K4–7 together with A^E_j ⊆ A^S_i imply (i, j) ∈ D.

In the instance of Figure 2.1 and Table 2.1, (3, 2) /∈ D because A^E₂ * A^S3 and B^E₃ * B2^S.

Given a dominant S ⊇ E, and a pair (i, j) such that S ∪ {(i, j)}

is acyclic, we define

Γ(S, i, j) = {(k, l) ∈ N ×N |k ∈ (B_i^E\B_j^S)∪{i}, l ∈ (A^E_j \A^S_i )∪{j}}

(2.3) as the set of all transitive pairs associated with (i, j) that are not yet included in S.

Proposition 2.7. If Γ(S, i, j) ⊆ C(S) then (i, j) ∈ D.

Proof. Proof. Given a set X of pairs, let X⁺be the transitive closure of X. Assume that we add the pairs in Γ(S, i, j) to S, sequentially.

We will do this in a specific order. If Γ(S, i, j) ⊆ C(S) then in the first step, there exist (k₁, l₁) ∈ Γ(S, i, j) and S₁ = S ∪ {(k₁, l₁)} such

18 2.2. PRECEDENCE THEOREMS

that S₁⁺∩ (Γ(S, i, j) \ S₁) = ∅. In other words, adding (k₁, l₁) to S does not imply any transitive pairs within Γ(S, i, j). For such a pair (k₁, l₁) we have A^E_l be verified to be dominant and S ∪ {(i, j)} is a dominant set.

2.2.3 Algorithmic application of the precedence the-orems

In this section, we illustrate the algorithmic application of Propos-ition 2.7. Given a dominant partial order S that extends E, we propose a framework, Frame1, for evaluating the dominance of a given pair (i, j) without generating C(S) explicitly.

The idea of Frame1 is to sequentially add the pairs in Γ(S, i, j) to S such that each addition entails no transitive pair within Γ(S, i, j) (as in the proof of Proposition 2.7). To this end, for each (k, l) ∈ Γ(S, i, j) we determine the longest path between k and l in the transitive reduction of the graph G(N, S ∪ {(i, j)}). We assume unit length (weight) for all the edges (activity pairs). The Floyd-Warshall algorithm, for instance, can be used to calculate these longest paths efficiently. We define L(S, i, j) as a sequence of the pairs in Γ(S, i, j) in non-increasing order of their corresponding longest path length.

Next, Frame1 checks the pairs in L(S, i, j) sequentially. Let (k_q, l_q) be the q^th element of L(S, i, j). In the first step, we check (k₁, l₁): if I(S, k₁, l₁) = 1 then we define S₁ = S ∪ {(k₁, l₁)} and we proceed to the next step; otherwise, we terminate the framework by concluding that (i, j) /∈ D. Analogously, in any step q > 1, if

CHAPTER 2. THE SINGLE-MACHINE WEIGHTED

Figure 2.5: An example of the application of Frame1

I(S_q−1, k_q, l_q) = 1 then we construct S_q = S_q−1∪ {(k_q, l_q)}. Note that by considering the pairs in Γ(S, i, j) in the order of L(S, i, j), based on Proposition 2.1, we benefit most from possible improve-ments in the theorem conditions for identifying the dominance of (k_q, lq). If all the activity pairs in L(S, i, j) are successfully added to S, then the framework ends with a dominant S⁰⊃ E that includes (i, j).

Figure 2.5 provides an illustration of the application of Frame1 to the instance depicted in Figure 2.1. At each step, the dashed arc represents the pair whose dominance is being evaluated, while the dotted arcs are the added pairs in the previous steps. For this example we have L(E, 3, 2) = ((1, 4), (1, 2), (3, 4), (3, 2)) for the corresponding longest path lengths 3, 2, 2 and 1 in the transitive reduction of G(N, E ∪ {(3, 2)}). Given the parameters in Figure 2.1, the framework terminates after the test for (1, 4) as I(E, 1, 4) = 0, by concluding that (3, 2) /∈ D. For other parameter values, if (3, 2) ∈ D then the framework iteratively adds further pairs ((1, 2), (3, 4), and (3, 2)) as depicted in Figures 2.5.